In [1]:
from google.colab import drive
drive.mount('/content/drive')
Drive already mounted at /content/drive; to attempt to forcibly remount, call drive.mount("/content/drive", force_remount=True).
In [2]:
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import warnings
warnings.filterwarnings('ignore')
from IPython.core.interactiveshell import InteractiveShell
InteractiveShell.ast_node_interactivity = "all"
%config InlineBackend.figure_format = 'svg'
%matplotlib inline
pd.set_option('expand_frame_repr', True)
pd.set_option('display.unicode.ambiguous_as_wide', True)
pd.set_option('display.unicode.east_asian_width', True)
pd.set_option('display.width', 180)
In [3]:
class BinomialModelAmericanPut:
def __init__(self, S0, K, T, r, sigma, mu, N):
self.S0 = S0 # Initial stock price
self.K = K # Strike price
self.T = T # Time to maturity
self.r = r # Risk-free rate
self.sigma = sigma # Volatility
self.mu = mu # Expected return
self.N = N # Number of steps
self.dt = T / N # Time step size
self.u = np.exp(r * self.dt + sigma * np.sqrt(self.dt)) # Up factor
self.d = np.exp(r * self.dt - sigma * np.sqrt(self.dt)) # Down factor
# Risk-neutral probabilities
self.p = (np.exp(r * self.dt) - self.d) / (self.u - self.d)
self.q = 1 - self.p
# Practical probabilities
self.real_p = (1 + (self.mu - self.r - self.sigma ** 2 / 2) * np.sqrt(self.dt) / self.sigma) / 2
self.real_q = 1 - self.real_p
# In the trees below, the i stands for th time step and the j stands for the state
# Initialize the option value tree and delta tree
self.option_values = np.full((N + 1, N + 1), np.nan)
self.delta_values = np.full((N + 1, N), np.nan)
# Initialize the exercise status tree
self.exercise_status = np.full((N + 1, N), np.nan)
self.realized_exercise = np.full((N + 1, N), np.nan)
# Initialize the stock price tree
self.stock_prices = np.full((N + 1, N + 1), np.nan)
self._build_stock_price_tree()
# Calculate option values using backward induction
self._calculate_option_values()
def _build_stock_price_tree(self):
# Build the stock price tree
# For each cell in the triangular matrix, (i - j) is the multiple of Up's and (j) is the multiple of Down's
for i in range(self.N + 1):
for j in range(i + 1):
self.stock_prices[j, i] = self.S0 * (self.u ** (i - j)) * (self.d ** j)
def _calculate_option_values(self):
# Initialize the option values at maturity
for j in range(self.N + 1):
self.option_values[j, self.N] = max(0, self.K - self.stock_prices[j, self.N])
# Backward induction to calculate option values and deltas
for i in range(self.N - 1, -1, -1):
for j in range(i + 1):
# Calculate the value if we hold the option
hold_value = np.exp(-self.r * self.dt) * (self.p * self.option_values[j, i + 1] +
self.q * self.option_values[j + 1, i + 1])
# Calculate the value if we exercise the option
exercise_value = self.K - self.stock_prices[j, i]
# Determine the exercise status for this step
if exercise_value >= hold_value:
self.exercise_status[j, i] = 0
self.option_values[j, i] = exercise_value
else:
self.exercise_status[j, i] = 1
self.option_values[j, i] = hold_value
# Calculate the delta
self.delta_values[j, i] = (self.option_values[j, i + 1] - self.option_values[j + 1, i + 1]) / (
self.stock_prices[j, i + 1] - self.stock_prices[j + 1, i + 1])
self.realized_exercise = self.exercise_status
def get_early_exercise_prices(self):
"""
Returns the stock prices at time step `t` where early exercise is optimal.
"""
self.exercise_prices_js = np.nansum(self.exercise_status, axis=0).astype(int)
self.exercise_prices = self.stock_prices[self.exercise_prices_js, np.arange(len(self.exercise_prices_js))]
return self.exercise_prices_js, self.exercise_prices
def fetch_along_the_path(self, path):
"""
Given a path (a list of 1s and 0s, 1 for a Down and 0 for an Up), return the values we concern along the path.
"""
js_path = np.insert(np.array(path), 0, 0).cumsum().astype(int)
prices_path = self.stock_prices[js_path, np.arange(len(js_path))]
# values_path = self.option_values[js_path, np.arange(len(js_path))]
# exercise_path = self.exercise_status[js_path[:-1], np.arange(len(js_path)-1)]
delta_path = self.delta_values[js_path[:-1], np.arange(len(js_path)-1)]
return prices_path, delta_path
def fetch_path_result(self, path):
"""
Given a path (a list of 1s and 0s, 1 for a Down and 0 for an Up), return the PnL and stopping time of the path.
"""
js_path = np.insert(np.array(path), 0, 0).cumsum().astype(int)
exercise_path = self.realized_exercise[js_path[:-1], np.arange(len(js_path)-1)]
if len(np.where(exercise_path == 0)[0]) > 0:
stopping_time = np.where(exercise_path == 0)[0][0]
PnL = self.K - self.stock_prices[js_path[stopping_time], stopping_time]
else:
stopping_time = len(js_path) - 1
PnL = self.option_values[js_path[stopping_time], stopping_time]
return PnL, stopping_time
def gen_simulation_paths(self, num_paths):
"""
Generate num_paths paths of the binomial tree.
"""
return np.random.choice([0, 1], size=(num_paths, self.N), p=[self.real_q, self.real_p])
def override_realized_exercise(self, exercise_boundary):
first_constrain = np.where(~np.isnan(exercise_boundary))[0][0]
for i in range(first_constrain):
for j in range(i+1):
self.realized_exercise[j, i] = 1
self.realized_exercise[self.stock_prices[:,:-1] > exercise_boundary] = 1
self.realized_exercise[self.stock_prices[:,:-1] <= exercise_boundary] = 0
return self.realized_exercise
In [4]:
S0 = 10 # Initial stock price
K = 10 # Strike price
T = 1 # Time to maturity (1 year)
r = 0.02 # Risk-free rate
sigma = 0.20 # Volatility
mu = 0.05 # Expected return
N = 5000 # Number of steps
tree = BinomialModelAmericanPut(S0, K, T, r, sigma, mu, N)
early_exercise_js, exercise_boundary = tree.get_early_exercise_prices()
first_constrain = np.nonzero(early_exercise_js - np.arange(1,N+1))[0][0]
# i)
# Plot the exercise boundary as a function of t
times = np.linspace(0, T, N + 1)
plt.figure(figsize=(8, 4))
plt.plot(times, np.append(exercise_boundary, np.nan), label="Exercise Boundary")
plt.xlim(0, T)
plt.xlabel("Time (t)")
plt.ylabel("Stock Price")
plt.title("Exercise Boundary for American Put")
plt.legend();
In [5]:
# ii)
# First generate the path where the option is exercised early near t = 1/2
# We need to check if it's possible to early exercise at int(N/2)
# Make sure that the boundary condition is not eliminated by former early exercises
t_half = int(N/2) - 1
while early_exercise_js[t_half] - early_exercise_js[t_half-1] == 1:
t_half += 1
# First we find the lower boundary path for the option to be early exercised at t_half, the path after t_half is generated randomly
path_list_half_lower_bound = np.concatenate((np.diff(np.append(early_exercise_js[0:t_half]-1, early_exercise_js[t_half])), np.random.randint(0, 2, size=N-t_half))).astype(int)
prices_path_half_lower_bound, delta_path_half_lower_bound = tree.fetch_along_the_path(path_list_half_lower_bound)
# We can also find the upper boundary path for the option to be early exercised at t_half, the path after t_half is generated randomly
path_list_half_upper_bound = np.concatenate((np.zeros(t_half-early_exercise_js[t_half]), np.ones(early_exercise_js[t_half]), np.random.randint(0, 2, size=N-t_half))).astype(int)
prices_path_half_upper_bound, delta_path_half_upper_bound = tree.fetch_along_the_path(path_list_half_upper_bound)
# All the paths in between fits our requirement
plt.figure(figsize=(8, 4))
times = np.linspace(0, T, N + 1)
plt.plot(times, prices_path_half_lower_bound, label=f'Lower Boundary Path (Early Exercise at t={t_half/N*T:.2f} years)')
plt.plot(times, prices_path_half_upper_bound, label=f'Upper Boundary Path (Early Exercise at t={t_half/N*T:.2f} years)')
plt.scatter(t_half/N*T, prices_path_half_lower_bound[t_half], color='red',label = 'Half Time Early Exercise Point')
plt.xlabel("Time (t)")
plt.ylabel("Stock Price")
plt.title("Sample Paths for American Put")
plt.legend();
In [6]:
# iii)
plt.figure(figsize=(8, 4))
times = np.linspace(0, T, N + 1)
plt.plot(times, np.append(delta_path_half_lower_bound, np.nan), label=f'Lower Boundary Path (Early Exercise at t={t_half/N*T:.2f} years)')
plt.plot(times, np.append(delta_path_half_upper_bound, np.nan), label=f'Upper Boundary Path (Early Exercise at t={t_half/N*T:.2f} years)')
plt.xlabel("Time (t)")
plt.ylabel("Hedging Delta")
plt.title("Hedging Delta for American Put")
plt.legend();
In [7]:
# Now generate the path where the option is not early exercised
t_full = N
# As the upper boundary path is trivial, we only look at the lower boundary path
path_list_full_lower_bound = np.append(np.diff(early_exercise_js[0:t_full+1]-1),1)
prices_path_full_lower_bound, delta_path_full_lower_bound = tree.fetch_along_the_path(path_list_full_lower_bound)
plt.figure(figsize=(8, 4))
times = np.linspace(0, T, N + 1)
plt.plot(times, prices_path_full_lower_bound, label='Lower Boundary Path (No Early Exercise)')
plt.xlabel("Time (t)")
plt.ylabel("Stock Price")
plt.title("Sample Paths for American Put")
plt.legend();
In [8]:
plt.figure(figsize=(8, 4))
times = np.linspace(0, T, N + 1)
plt.plot(times, np.append(delta_path_full_lower_bound, np.nan), label='Lower Boundary Path (No Early Exercise)')
plt.xlabel("Time (t)")
plt.ylabel("Hedging Delta")
plt.title("Hedging Delta for American Put")
plt.legend();
In [9]:
# iv)
for sigma_new in [0.10, 0.20, 0.30]:
for r_new in [0.00, 0.02, 0.04]:
tree = BinomialModelAmericanPut(S0, K, T, r_new, sigma_new, mu, N)
early_exercise_js, exercise_boundary = tree.get_early_exercise_prices()
# Plot the exercise boundary as a function of t
times = np.linspace(0, T, N + 1)
plt.figure(figsize=(8, 4))
plt.plot(times, np.append(exercise_boundary, np.nan), label="Exercise Boundary")
plt.xlim(0, T)
plt.xlabel("Time (t)")
plt.ylabel("Stock Price")
plt.title(f"Exercise Boundary for American Put with sigma = {sigma_new}; r = {r_new}")
plt.legend()
# plt.savefig(f'/content/drive/My Drive/SyncWorkspace/MMF/MMF1928H/Project 1/Exercise_boundary_plot-sigma={sigma_new}&r={r_new}.jpg')
# Generate the path where the option is exercised early near t = 1/2
t_half = int(N/2) - 1
while early_exercise_js[t_half] - early_exercise_js[t_half-1] == 1:
t_half += 1
path_list_half_lower_bound = np.concatenate((np.diff(np.append(early_exercise_js[0:t_half]-1, early_exercise_js[t_half])), np.random.randint(0, 2, size=N-t_half))).astype(int)
prices_path_half_lower_bound, delta_path_half_lower_bound = tree.fetch_along_the_path(path_list_half_lower_bound)
# Now generate the path where the option is not early exercised
t_full = N
path_list_full_lower_bound = np.append(np.diff(early_exercise_js[0:t_full+1]-1),1)
prices_path_full_lower_bound, delta_path_full_lower_bound = tree.fetch_along_the_path(path_list_full_lower_bound)
# Plot the price paths
plt.figure(figsize=(8, 4))
times = np.linspace(0, T, N + 1)
plt.plot(times, prices_path_half_lower_bound, label=f'Lower Boundary Path (Early Exercise at t={t_half/N*T:.2f} years)')
plt.scatter(t_half/N*T, prices_path_half_lower_bound[t_half], color='red',label = 'Half Time Early Exercise Point')
plt.plot(times, prices_path_full_lower_bound, label='Lower Boundary Path (No Early Exercise)')
plt.xlabel("Time (t)")
plt.ylabel("Stock Price")
plt.title(f"Sample Paths for American Put with sigma = {sigma_new}; r = {r_new}")
plt.legend()
# plt.savefig(f'/content/drive/My Drive/SyncWorkspace/MMF/MMF1928H/Project 1/Sample_prices_path_plot-sigma={sigma_new}&r={r_new}.jpg')
# Plot the hedging delta
plt.figure(figsize=(8, 4))
times = np.linspace(0, T, N + 1)
plt.plot(times, np.append(delta_path_half_lower_bound, np.nan), label=f'Lower Boundary Path (Early Exercise at t={t_half/N*T:.2f} years)')
plt.plot(times, np.append(delta_path_full_lower_bound, np.nan), label='Lower Boundary Path (No Early Exercise)')
plt.xlabel("Time (t)")
plt.ylabel("Hedging Delta")
plt.title(f"Hedging Delta for American Put with sigma = {sigma_new}; r = {r_new}")
plt.legend()
# plt.savefig(f'/content/drive/My Drive/SyncWorkspace/MMF/MMF1928H/Project 1/Delta_path_plot-sigma={sigma_new}&r={r_new}.jpg')
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In [10]:
# pd.DataFrame(tree.stock_prices).to_csv('/content/drive/My Drive/SyncWorkspace/MMF/MMF1928H/Project 1/stock_prices.csv')
# pd.DataFrame(tree.option_values).to_csv('/content/drive/My Drive/SyncWorkspace/MMF/MMF1928H/Project 1/option_values.csv')
# pd.DataFrame(tree.delta_values).to_csv('/content/drive/My Drive/SyncWorkspace/MMF/MMF1928H/Project 1/delta_values.csv')
# pd.DataFrame(tree.exercise_status).to_csv('/content/drive/My Drive/SyncWorkspace/MMF/MMF1928H/Project 1/exercise_status.csv')
In [11]:
# (b)
# i)
S0 = 10 # Initial stock price
K = 10 # Strike price
T = 1 # Time to maturity (1 year)
r = 0.02 # Risk-free rate
sigma = 0.20 # Volatility
mu = 0.05 # Expected return
N = 5000 # Number of steps
tree = BinomialModelAmericanPut(S0, K, T, r, sigma, mu, N)
# Simulate the paths
sample_paths = tree.gen_simulation_paths(10000)
# # Use the `path_list_half_lower_bound` and `path_list_full_lower_bound` before as examples
# PnL_early_exercise, stopping_time_early_exercise = tree.fetch_path_result(path_list_half_lower_bound)
# PnL_early_exercise, stopping_time_early_exercise
# PnL_no_early_exercise, stopping_time_no_early_exercise = tree.fetch_path_result(path_list_full_lower_bound)
# PnL_no_early_exercise, stopping_time_no_early_exercise
In [12]:
# Get the PnL and stopping time for each path
PnL_distirbution = []
stopping_time_distribution = []
for path in sample_paths:
PnL_early_exercise, stopping_time_early_exercise = tree.fetch_path_result(path)
PnL_distirbution.append(PnL_early_exercise)
stopping_time_distribution.append(stopping_time_early_exercise / tree.N)
In [13]:
# Plot the PnL histogram
PnL_continuous_part = [x for x in PnL_distirbution if x > 0] # All values less than 1
PnL_point_mass_count = len([x for x in PnL_distirbution if x == 0]) # Count the number of occurrences of 1
# Plot the histogram for the continuous part
plt.figure(figsize=(8, 6))
plt.hist(PnL_continuous_part, bins=50, density=True, alpha=0.7, color='blue', label='Continuous Part')
# Add the point mass at 1
plt.axvline(x=0, color='red', linestyle='--', linewidth=8, label='Point Mass at 0')
# Annotate the point mass with its relative probability
PnL_relative_mass_prob = PnL_point_mass_count / len(PnL_distirbution) # The probability mass at 1
plt.text(0., -0.05, f'Point Mass: {PnL_relative_mass_prob:.2f}', color='red')
plt.title("PnL Distribution with Point Mass at 0", fontsize=15)
plt.xlabel("Value", fontsize=12)
plt.ylabel("Density", fontsize=12)
plt.legend()
plt.grid(True);
In [14]:
# Separate the continuous part and the point mass part
stopping_time_continuous_part = [x for x in stopping_time_distribution if x < 1] # All values less than 1
stopping_time_point_mass_count = len([x for x in stopping_time_distribution if x == 1]) # Count the number of occurrences of 1
# Plot the histogram for the continuous part
plt.figure(figsize=(8, 6))
plt.hist(stopping_time_continuous_part, bins=50, density=True, alpha=0.7, color='blue', label='Continuous Part')
# Add the point mass at 1
plt.axvline(x=1, color='red', linestyle='--', linewidth=2, label='Point Mass at 1')
# Annotate the point mass with its relative probability
stopping_time_relative_mass_prob = stopping_time_point_mass_count / len(stopping_time_distribution) # The probability mass at 1
plt.text(1+0.01, 0., f'Point Mass: {stopping_time_relative_mass_prob:.2f}', color='red')
plt.title("Stopping Time Distribution with Point Mass at 1", fontsize=15)
plt.xlabel("Value", fontsize=12)
plt.ylabel("Density", fontsize=12)
plt.legend()
plt.grid(True);
In [15]:
# ii)
for sigma_new in [0.10, 0.20, 0.30]:
for r_new in [0.00, 0.02, 0.04]:
tree = BinomialModelAmericanPut(S0, K, T, r_new, sigma_new, mu, N)
sample_paths = tree.gen_simulation_paths(10000)
PnL_distirbution = []
stopping_time_distribution = []
for path in sample_paths:
PnL_early_exercise, stopping_time_early_exercise = tree.fetch_path_result(path)
PnL_distirbution.append(PnL_early_exercise)
stopping_time_distribution.append(stopping_time_early_exercise / tree.N)
# Plot the PnL histogram
PnL_continuous_part = [x for x in PnL_distirbution if x > 0]
PnL_point_mass_count = len([x for x in PnL_distirbution if x == 0])
plt.figure(figsize=(8, 6))
plt.hist(PnL_continuous_part, bins=50, density=True, alpha=0.7, color='blue', label='Continuous Part')
plt.axvline(x=0, color='red', linestyle='--', linewidth=8, label='Point Mass at 0')
PnL_relative_mass_prob = PnL_point_mass_count / len(PnL_distirbution)
plt.text(0., -0.05, f'Point Mass: {PnL_relative_mass_prob:.2f}', color='red')
plt.title(f"PnL Distribution of sigma = {sigma_new}; r = {r_new} with Point Mass at 0", fontsize=15)
plt.xlabel("Value", fontsize=12)
plt.ylabel("Density", fontsize=12)
plt.legend()
plt.grid(True)
# plt.savefig(f'/content/drive/My Drive/SyncWorkspace/MMF/MMF1928H/Project 1/PnL_distribution_plot-sigma={sigma_new}&r={r_new}.jpg')
# Separate the continuous part and the point mass part
stopping_time_continuous_part = [x for x in stopping_time_distribution if x < 1]
stopping_time_point_mass_count = len([x for x in stopping_time_distribution if x == 1])
plt.figure(figsize=(8, 6))
plt.hist(stopping_time_continuous_part, bins=50, density=True, alpha=0.7, color='blue', label='Continuous Part')
plt.axvline(x=1, color='red', linestyle='--', linewidth=2, label='Point Mass at 1')
stopping_time_relative_mass_prob = stopping_time_point_mass_count / len(stopping_time_distribution)
plt.text(1+0.01, 0., f'Point Mass: {stopping_time_relative_mass_prob:.2f}', color='red')
plt.title(f"Stopping Time Distribution of sigma = {sigma_new}; r = {r_new} with Point Mass at 1", fontsize=15)
plt.xlabel("Value", fontsize=12)
plt.ylabel("Density", fontsize=12)
plt.legend()
plt.grid(True)
# plt.savefig(f'/content/drive/My Drive/SyncWorkspace/MMF/MMF1928H/Project 1/Stopping_time_distribution_plot-sigma={sigma_new}&r={r_new}.jpg')
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In [16]:
# iii)
# First we get the exercise boundary of the Put assuming sigma = 0.20
S0 = 10 # Initial stock price
K = 10 # Strike price
T = 1 # Time to maturity (1 year)
r = 0.02 # Risk-free rate
sigma = 0.20 # Volatility
mu = 0.05 # Expected return
N = 5000 # Number of steps
tree = BinomialModelAmericanPut(S0, K, T, r, sigma, mu, N)
early_exercise_js, exercise_boundary = tree.get_early_exercise_prices()
# Then we calculate the price paths for all possible realized volatilities
for sigma_new in [0.10]:#, 0.15, 0.20, 0.25, 0.30]:
tree = BinomialModelAmericanPut(S0, K, T, r, sigma_new, mu, N)
realized_exercise = tree.override_realized_exercise(exercise_boundary)
sample_paths = tree.gen_simulation_paths(10000)
PnL_distirbution = []
stopping_time_distribution = []
for path in sample_paths:
PnL_early_exercise, stopping_time_early_exercise = tree.fetch_path_result(path)
PnL_distirbution.append(PnL_early_exercise)
stopping_time_distribution.append(stopping_time_early_exercise / tree.N)
# Plot the PnL histogram
PnL_continuous_part = [x for x in PnL_distirbution if x > 0]
PnL_point_mass_count = len([x for x in PnL_distirbution if x == 0])
plt.figure(figsize=(8, 6))
plt.hist(PnL_continuous_part, bins=50, density=True, alpha=0.7, color='blue', label='Continuous Part')
plt.axvline(x=0, color='red', linestyle='--', linewidth=8, label='Point Mass at 0')
PnL_relative_mass_prob = PnL_point_mass_count / len(PnL_distirbution)
plt.text(-0.55, 0., f'Point Mass: {PnL_relative_mass_prob:.2f}', color='red')
plt.title(f"PnL Distribution of Miss-matched realized sigma = {sigma_new} with Point Mass at 0", fontsize=15)
plt.xlabel("Value", fontsize=12)
plt.ylabel("Density", fontsize=12)
plt.legend()
plt.grid(True)
# plt.savefig(f'/content/drive/My Drive/SyncWorkspace/MMF/MMF1928H/Project 1/Miss-match-vol_PnL_distribution_plot-sigma={sigma_new}.jpg')
# Separate the continuous part and the point mass part
stopping_time_continuous_part = [x for x in stopping_time_distribution if x < 1]
stopping_time_point_mass_count = len([x for x in stopping_time_distribution if x == 1])
plt.figure(figsize=(8, 6))
plt.hist(stopping_time_continuous_part, bins=50, density=True, alpha=0.7, color='blue', label='Continuous Part')
plt.axvline(x=1, color='red', linestyle='--', linewidth=2, label='Point Mass at 1')
stopping_time_relative_mass_prob = stopping_time_point_mass_count / len(stopping_time_distribution)
plt.text(1+0.01, 0., f'Point Mass: {stopping_time_relative_mass_prob:.2f}', color='red')
plt.title(f"Stopping Time Distribution of Miss-matched realized sigma = {sigma_new} with Point Mass at 1", fontsize=15)
plt.xlabel("Value", fontsize=12)
plt.ylabel("Density", fontsize=12)
plt.legend()
plt.grid(True)
# plt.savefig(f'/content/drive/My Drive/SyncWorkspace/MMF/MMF1928H/Project 1/Miss-match-vol_Stopping_time_distribution_plot-sigma={sigma_new}.jpg')
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